zbMATH — the first resource for mathematics

Diffusion and memory effects for stochastic processes and fractional Langevin equations. (English) Zbl 1050.82029
Summary: We consider the diffusion processes defined by stochastic differential equations when the noise is correlated. A functional method based on the Dyson expansion for the evolution operator, associated to the stochastic continuity equation, is proposed to obtain the Fokker-Planck equation, after averaging over the stochastic process. In the white noise limit the standard result, corresponding to the Stratonovich interpretation of the nonlinear Langevin equation, is recovered. When the noise is correlated the averaged operator series cannot be summed, unless a family of time-dependent operators commutes. In the case of a linear equation, the constraints are easily worked out. The process defined by a linear Langevin equation with additive noise is Gaussian and the probability density function of its fluctuating component satisfies a Fokker-Planck equation with a time-dependent diffusion coefficient. The same result holds for a linear Langevin equation with a fractional time derivative [defined according to M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna (1969)]. In the generic linear or nonlinear case approximate equations for small noise amplitude are obtained. For small correlation time the evolution equations further simplify in agreement with some previous alternative derivations. The results are illustrated by the linear oscillator with coloured noise and the fractional Wiener process, where the numerical simulation for the probability density and its moments is compared with the analytical solution.

82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
[1] M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969.
[2] Dyson, F., The radiation theories of Tomonaga, Schwinger and feynmann, Phys. rev., 75, 486-502, (1949) · Zbl 0032.23702
[3] Young, P., (), 23-31
[4] Graham, R., Covariant formulation of non equilibrium statistical thermodynamics, Z. phys., 26, 397-405, (1977)
[5] Van Kampen, N.G., Stochastic differential equations, Phys. rep., 24, 171-228, (1976)
[6] Kubo, R., (), 23
[7] Risken, H., The fokker – planck equation, (1984), Springer Berlin · Zbl 0546.60084
[8] Hänggi, P.; Jung, P., Coloured noise in dynamical systems, Adv. chem. phys., 89, 238-318, (1995), and references therein
[9] Heinrichs, J., Probability distribution for second order processes driven by Gaussian noise, Phys. rev. E, 47, 3007-3012, (1993)
[10] Faetti, S.; Grigolini, P., Unitary point of view in the puzzling problem of nonlinear systems driven by coloured noise, Phys. rev. A, 36, 441-444, (1987)
[11] Sancho, J.M.; San Miguel, M., External nonwhite noise and non equilibrium phase transitions, Z. phys. B, 36, 357-364, (1980)
[12] Ferrario, M.; Grigolini, P., The non markovian relaxation process as a contraction of a multidimensional one of Markovian type, J. math. phys., 20, 2587, (1978)
[13] Fox, R.F., Gaussian stochastic processes in physics, Phys. rep., 48, 179-283, (1978)
[14] Bazzani, A.; Siboni, S.; Turchetti, G., Diffusion in Hamiltonian systems with a small stochastic perturbation, Physica D, 76, 8, (1994) · Zbl 1194.34117
[15] Khas’minskii, Z.R., On stochastic processes defined by differential equations with a small parameter, Theory probab. appl., 11, 211-228, (1966) · Zbl 0168.16002
[16] Cogburn, R.; Ellison, J.A., A stochastic theory of adiabatic invariance, Commun. math. phys., 149, 97-126, (1992) · Zbl 0755.60095
[17] Bazzani, A.; Beccaceci, L., Diffusion in Hamiltonian systems with a coloured noise, J. phys. A, 39, 139, (1999)
[18] Kubo, R., Stochastic Liouville equations, J. math. phys., 4, 174-183, (1963) · Zbl 0135.45102
[19] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II, Nonequilibrium statistical mechanics, Springer Verlag Series in Solid State Science, Vol. 31, Springer, Berlin, 1978, p. 25. · Zbl 0996.60501
[20] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. calculus appl. anal., 4, 153-192, (2001) · Zbl 1054.35156
[21] Turchetti, G.; Usero, D.; Vazquez, L., Hamiltonian systems with fractional time derivative, Tamsui Oxford J. math. sci., 18, 31-44, (2002) · Zbl 1020.37043
[22] Kobelev, V.; Romanov, E., Fractional Langevin equation to describe anomalous diffusion, Prog. theoret. phys., 139, Suppl., 470-476, (2000)
[23] E. Lutz, Fractional Langevin equation, preprint in electronic form at arXiv: cond-mat/0103128, 2001.
[24] F. Mainardi, P. Paradisi, R. Gorenflo, Probability distributions as solutions to fractional diffusion equations, Ma-Phy-Sto Conference on Levy Processes Theory and Applications, University of Aarhus, January 2002, 197-205, MPS-misc 2002-22 downloadable from www.fracalmo.org.
[25] Gorenflo, R.; Mainardi, F.; Moretti, D.; Pagnini, G.; Paradisi, P., Discrete random walk models for space time fractional diffusion, Chem. phys., 284, 521-541, (2002)
[26] Chechkin, A.V.; Yu Gonchar, V., Linear relaxation governed by fractional symmetric kinetic equations, Tom, 118, 730-748, (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.